Randomly coloring constant degree graphs1. Issue 2 (3rd August 2012) - Dyer, Martin; Frieze, Alan; Hayes, Thomas P.; Vigoda, Eric
Abstract We study a simple Markov chain, known as the Glauber dynamics, for generating a random k ‐coloring of an n ‐vertex graph with maximum degree Δ. We prove that, for every ε > 0, the dynamics converges to a random coloring within O(nlog n) steps assuming k ≥ k0(ε) and either: (i) k/Δ > α* + ε where α*≈︁ 1.763 and the girth g ≥ 5, or (ii) k/Δ >β * + ε where β*≈︁ 1.489 and the girth g ≥ 7. Our work improves upon, and builds on, previous results which have similar restrictions on k/Δ and the minimum girth but also required Δ = Ω (log n). The best known result for general graphs is O(nlog n) mixing time when k/Δ > 2 and O(n2) mixing time when k/Δ > 11/6. Related results of Goldberg et al apply when k/Δ > α* for all Δ ≥ 3 on triangle‐free "neighborhood‐amenable" graphs.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013